The name of the number is …

The interparietal sulcus also shows up in a New York Times article (December 20, 2009): “Studying Young Minds, and How to Teach Them.” There Benedict Carey explains how “findings, mostly from a branch of research called cognitive neuroscience, are helping to clarify when young brains are best able to grasp fundamental concepts.” He quotes Kurt Fisher, director of the Mind, Brain and Education program at Harvard: “… for the first time we are seeing the fields of brain science and education work together.” In mathematics, this means starting early to develop children’s innate apprehension of number (see previous item) into the precise tool they need to succeed in kindergarten and beyond. “By preschool, the brain can handle larger numbers and is struggling to link three crucial concepts: physical quantities (seven marbles, seven inches) with abstract digit symbols (“7”), with the corresponding number words (“seven” ).” To show us how this works, Carey takes us into a classroom in Buffalo where Mrs. Pat Andzel is leading her preschoolers, over and over in different and often entertaining contexts, through the algorithm of counting: the name of the number of things in some set is the last word you pronounce when you count them.“Many of these kids don’t understand that yet,” she says.

I feel an instinctive disagreement with this “algorithm”; I bet, as a child, I would not understand it. And what are you feelings?

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Yeah, I don’t like it either. I’m a teacher at a secondary school in Sweden and it seems to me that teaching concepts through algorithms must be popular at lower levels. Kids like algorithms because they can just memorize them and teachers like them (I’m speculating) because they’re easy to teach, for the same reason. In algebra, for instance, they learn to “cross multiply” or “move” terms around in an equation. That type of algorithmic thinking shortcut is useful and important, but only if you first understand what it actually means: that you can maintain the truth value of an equation if you apply the same operation to both sides. Upon arrival at our school almost none of our kids have any idea of what an equation is, or why some operations are allowed and not others, because they’ve only ever learned the algorithms. The problem with this is that they have no ability to do a sanity check on a misremembered algorithm. For instance, “moving” a term means adding or subtracting on both sides, but if you know the algorithm as “move”, you’re likely to forget that the operation affects the sign of the term. An even bigger problem than the potential for mistakes is that they never learn to think about mathematics except as problems to which they apply their algorithms. That approach rarely works outside of a classroom, and not always there. Algorithms are clearly useful and necessary, and may seem concrete, but are actually very abstract as a mode of understanding.

The algorithm of naming by counting strikes me as bizarre. I don’t work with younger children and only have my own to go by, but is naming a concept really the hard part? Naming things, even concepts, is such a natural part of language development. I can’t help feeling that the complicated rule they’re teaching kids is a barrier to actual intuitive understanding, rather than an aid. Instead, just have them spend a lot of time playing with numbers in different contexts. They’ll get it. It’s what humans do.

After discussing this with a colleague I feel like I misunderstood the approach cited when I wrote my last comment. It really isn’t as bizarre as it first seemed to me. Many children first learn counting as a sort of sequence of words, then as an algorithm where you point to each item in a set of objects and say a word in the sequence. If you run out of objects at the word “seven” then there are seven objects. It’s only a strange way of teaching if you try to actually explain this to the kids. And after thinking about it, I’m pretty sure that’s not what they’re doing. They’re counting seven blocks and then saying, “Look, seven blocks!”. That sounds to me like it could be useful.

Different children learn about numbers in different ways. My elder son was very slow in learning to “count”, meaning “recite numbers one, two, three… in order”. He was still refusing to go beyond five or six when the kid up the road, three weeks older, could reach twenty. But when, around that time, I was building Lego models for him, and wanted him to pick out twelve pieces for the next stage, I could ask “three and three, and three and three” and get them. I remember also watching him build a rectangular corral out of his bricks, asking “how many do you need to finish it”, and getting the (correct) answer “three of these and one of these”.

When we came to arithmetic, neither of my boys had a moment’s trouble; and I think that in the case of the elder, at least, he knew the number thirteen (for instance) before he knew its name.

Sometimes when travelling, our three kids would pick a digit from 1 to 9 and keep a running total of the number of times they saw their digit in a sign, etc. Soon our youngest son asked if he could change to two. We said yes. And Jon started calling out two, two two, … Jon where are you seeing all of those 2’s?. “Easy, two cows, two telephone poles, two cars.”